There is a difference between "Smart Rotation" which rotates the screen based on the orientation of your face and "Auto Rotate Screen" which orients the screen based on how you are holding the device (landscape vs. portrait). It is the "Smart Rotation" that gives you the little "eye" icon on the task bar.

And as to why the icon sometimes appears and sometimes doesn't (or perhaps is sometimes grayed out and sometimes white), I believe that relates to whether the current app is able to use this feature: if not, the icon disappears or is grayed out. I think.

The comment made me smile. The same can be said for nearly every app.
I personally haven't noticed much of a drain on my battery. It's only in use when you are actively using the phone, otherwise it's sleeping.

Is it supposed to stay white like all the other icons in the taskbar? For the most part mine is grayed out but I notice it turning white on occasion. Does that mean anything?

When it is grey that means it's not looking at your eye through the camera, but when it flashes white that means it's using the camera to look at your eye. It will only flash white when it is near your screen timeout. For example, life your screen tike out is 30 seconds, when the display has been left on and untouched for 25ish seconds the eye will be greyed out and the camera is being unused, but after 25 sec the icon will turn white and detect if you u are still looking at the screen

Not sure if I am talking about the same thing but when I am connected to the internet on my S4 I keep seeing a blue or green eye like icon which is similar but not quite the same as the one on the taskbar. It appears for a few seconds, sometimes switches colour the disappears.
How do I stop it?